Mathematical Model on the Transmission of Dengue Fever

Transcription

1 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: AUSTRALAN JOURNAL OF BASC AND APPLED SCENCES SSN: ESSN: Journal ome page: Matematical Model on te Transmission of Dengue Feer Kingdao Cangklang, Surapol Naowarat Prasit Tongjaem Department of Matematics, Faculty of Science Tecnology, Surattani Rajabat Uniersity,Surat Tani, 841, Tail. Address For Correspondence: Surapol Naowarart, Department of Matematics, Faculty of Science Tecnology, Surattani Rajabat Uniersity,Surat Tani, 841, Tail. A R T C L E N F O Article istory: Receied 3 Marc 216 Accepted 2 May 216 publised 26 May 216 Keywords: Matematic model, Dengue feer, Basic reproductie number Equilibrium points, Stability. A B S T R A C T Te objecties of tis study were to propose te matematical model on te transmission of Dengue feer to analyze te matematical model of Dengue feer transmission. Te uman population is diided into 3 subclass, tat is, te susceptible uman, infected uman recoered uman te mosquito population wic diided into 2 subclass, tat is, te susceptible mosquito infected mosquito. We inestigate te effect of parameter uman ) (te transmission rate from mosquito to - (te transmission rate from uman to mosquito) te calculation of te leel of infection (Basic Reproductie Number : R ) to study disease free equilibrium disease endemic equilibrium states. After tat, to determine te solutions by using te numerical simulation at eac equilibrium points wic affected to te local asymptotically stabilities, to determine of parameter (te transmission rate from mosquito to uman ) from mosquito to uman), by using =.5 (te transmission rate =.5 wic sowed R = < 1. Te results sowed tat te all subclass of uman were uninfected, it also sowed tat tere were no te transmission of te disease. n addition, as regards to te numerical simulation te parameter at disease free equilibrium wic affected to te local asymptotically stabilities, te results of te parameter sowed;, by using =. 9 =. 9 te results sowed R = > 1 tat mean tere are te transmission of te disease. NTRODUCTON Dengue is a mosquito-borne iral disease tat as rapidly spread in all regions of WHO in recent years. Dengue irus is transmitted by female mosquitoes mainly of te species Aedes aegypti, to a lesser extent, Ae. albopictus. Tis mosquito also transmits cikungunya, yellow feer Zika infection. Dengue is widespread trougout te tropics, wit local ariations in risk influenced by rainfall, temperature unplanned rapid urbanization. Seere dengue (also known as Dengue Haemorragic Feer) was first recognized in te 195s during dengue epidemics in te Pilippines Tail. Today, seere dengue affects most Asian Latin American countries as become a leading cause of ospitalization deat among cildren in tese regions. Tere Open Access Journal Publised BY AENS Publication 216 AENS Publiser All rigts resered Tis work is licensed under te Creatie Commons Attribution nternational License (CC BY). ttp://creatiecommons.org/licenses/by/4./ To Cite Tis Article: Kingdao Cangklang, Surapol Naowarat Prasit Tongjaem., Matematical Model on te Transmission of Dengue Feer. Aust. J. Basic & Appl. Sci., 1(11): , 216

2 15 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: are 4 distinct, but closely related, serotypes of te irus tat cause dengue (DEN-1, DEN-2, DEN-3 DEN-4). Recoery from infection by one proides lifelong immunity against tat particular serotype. Howeer, crossimmunity to te oter serotypes after recoery is only partial temporary. Subsequent infections by oter serotypes increase te risk of deeloping seere dengue. Te incidence of dengue as grown dramatically around te world in recent decades. Te actual numbers of dengue cases are underreported many cases are misclassified. One recent estimate indicates 39 million dengue infections per year (95% credible interal million), of wic 96 million ( million) manifest clinically (wit any seerity of disease).1 Anoter study, of te prealence of dengue, estimates tat 3.9 billion people, in 128 countries, are at risk of infection wit dengue iruses.(who, 216). Te Aedes aegypti mosquito is te primary ector of dengue. Te irus is transmitted to umans troug te bites of infected female mosquitoes. After irus incubation for 4 1 days, an infected mosquito is capable of transmitting te irus for te rest of its life. nfected symptomatic or asymptomatic umans are te main carriers multipliers of te irus, sering as a source of te irus for uninfected mosquitoes. Patients wo are already infected wit te dengue irus can transmit te infection (for 4 5 days; maximum 12) ia Aedes mosquitoes after teir first symptoms appear. (CDC,215). n tis paper, we proposed a matematical model to describe te transmission of Dengue feer. 2. Model Formulation: n our model, we assume tat te uman is constant. We formulate te model of Dengue feer transmission by using basic ideas taken from epidemiology tat is mosquito-borne is ig wic effect to Dengue feer epidemic. Te model is obtained by assuming: Te total uman population N is diided into four compartments: Susceptible uman denote by S (te members of te uman population wo may become infected). nfected mosquitoes denoted by ( te infected ) Recoer uman denoted by R (te numbers of recoered indiiduals). Te dynamics of te disease is depicted in te flow cart sown in Fig. 1. Fig. 1: Flow cart of te dynamics of dengue feer. Te dynamics of dengue is described by te following ordinary differential equations: ds = Λ S S d = S ( ) dr ds d wit were γ R = (3) = Λ S S = S (5) 1 + N = S + R + (1) (2) (4)

3 16 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: S is te susceptible umans, is te infectie umans, R is te recoered umans, Λ is te birt rate of umans, is te natural deat rate of umans, γ is te recoery rate of disease, is te infection rate from umans to mosquitoes, N is te total uman population, S is te susceptible mosquitoes, is te infected mosquitoes, is te natural deat rate of mosquitoes, is te infection rate from mosquitoes to umans, is te total number mosquitoes. N 3. Analysis of te Model: Equilibrium Points: Te system as two equilibrium points; a disease free equilibrium point an endemic equilibrium point. We obtained tese by setting te rigt sides of equations. (1) - (5) to zero. Doing tis, we obtained Disease Free Equilibrium Point: ( E ): n te absence of te disease, i.e., = ds dr = Λ = γ R S S Te solution to tis equation is, = equation (1), (2) becomes E = (1,, ) γ R =. Te disease free state is μ Endemic Equilibrium Point: ( E 1 ): n te case were te disease is presented, by setting,. Tis gies Tus E ( S,, 1 )

4 17 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: Basic Reproductie Number: Te basic reproductie number is obtained by te next generation matrix. n te notation of Van den Driessce Watmoug (22), we start wit dx = F( x) V( x) (6) were F (x) is te matrix of new infectious V(x) is te matrix of te transfers between te compartments in te infectie equations. We obtained S F( x) = S + S N V ( x) = ( ) ( Λ ) + ( ) were F = F i ( E ) V = i ( E V ) X i X i for all i, j = 1, 2, 3. Tis are te Jacobian matrix of F (x) V (x) at E. Te basic reproductie number, R, is te tresold for indicating te degree of epidemiology of te disease. t can be determined by noting tat 1 R = ρ( FV ) For our model, te Jacobian matrices are F = V = γ + V 1 FV -1 Te inerse of is V 1 ( ) 1 = 1 ( ) Tis leads to = ( )

5 18 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: Tus, R = ( ) (7) Local Asymptotically Stability: Te local stability of an equilibrium point is determined from te Jacobian matrix of te ordinary differential equation (1),(2) (5) ealuated at E. Te Jacobian matrix at E is - - J = - γ - Te eigenalues of te J det ( J λ) =. From tis, we obtain te caracteristic equation, [ ] ( λ + ) (-μ -γ -λ)(-μ -λ)- = 2 ( λ + μ )[ λ + Bλ + C] = were B = + C = ( ) From te caracteristic equation, we see tat two eigenalues are λ = μ. Te oter two are te 1 < solutions of te caracteristic equation Te roots of tis equation will be negatie if two coefficients satisfied wit te Rout-Hurwitz criteria (Allen,26). 1) B > 2) C > Disease Endemic Equilibrium Point: To determine te stability of te endemic equilibrium point. We examine te eigenalues of Jacobian matrix at E 1, wic is Were are gien by equations (6). Te caracteristic equation of Jacobian matrix at E1 gien by equations (1),(2) (5) becomes, R = 3 2 λ Pλ Qλ were u =, ( Λ ) 2 (1 + ) x = z = P = u w z Q = uz + wz + uw xy =, w = γ, 1 +, S y = (1 + ) 2,

6 19 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: R = xy + uxy uwz 3 2 λ λ λ Te tree eigenalues of Hurwitz criteria (Allen,26), tat is PQ > R. + P + Q + R = will ae negatie real part if tey satisfy te Rout - 4 Numerical Results: Te alue of parameters used in te numerical simulation are gien in Table 1. Table 1: Parameter alues used in numerical simulations at disease free state. Parameter Values Unit Λ.46 day day -1 γ day N 1, - Λ.323 day day N 5, - Stability of te disease free state: Using te alues of parameters listed in Table 1. We find te eigenalues basic reproductie number to be: λ 1 =.46, λ 2 =.2711, λ 3 =.35228, R = < Since all of te eigenalues are negatie te basic reproductie number is less tan one, te equilibrium state will be te disease free state E, as seen in Fig. 2 Fig. 2: Te time series of (a) te susceptible umans, (b) te infected umans (c) infected mosquitoes. As sown, all te state ariables approac to te disease free state as seen, all te state ariables approac teir disease free state alues E = (1,, ).

7 11 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: Fig. 3: Te time series of (a) te susceptible umans, (b) te infected umans (c) infected mosquitoes. Two te alue of as been canged to =. 9 =. 9. All te state ariables approac to endemic state. Stability of te endemic state: Using te alues of parameter listed in Table 1. except te alue of. We set to be equal.9. Tis alues represented te alue of infection from uman to mosquitoes te alue of infection from mosquitoes to umans. E 1 = ( ,.6819,.88781) Te eigenalues basic reproductie number become greater tan one te outcome is quite different: λ 1 =.27463, λ 2 = i λ 3 = i R = >1 Since all of te eigenalues are to be negatie te basic reproductie number is greater tan one, te equilibrium state will be te endemic state, E 1 as demonstrated in Fig. 3. Discussion Conclusion: n tis study, we proposed analyzed te transmission model of Dengue feer wit effect of te infection rate between uman mosquitoes. Model analysis by using stard modeling metod. Te basic reproductie number is obtained troug te use of spectral radius of te next generation matrix. Te basic reproductie number is R =. From Fig. 3, we see tat if te infection rate bot uman-to ( ) mosquito mosquito- to- uman increase, te number of infected uman increase. So tat te disease will occur. But wen te infection rate bot uman-to mosquito mosquito- to- uman decrease, te number of infected uman decrease. n tis case te disease will died out. t can summarize as follows: 1) Te spreading of dengue feer as two states: te disease-free state te endemic state. Te appening of a state depends on occur, but if =.9. f =.5 =.5 proided R < 1, ten te disease-free state will =.9 proided R > 1 ten te endemic state will occur. 2) Te stability of te model is determined. Rout-Hurwitz criteria is used to proe tat eac equilibrium point is locally asymptotically stable. n addition, for any initial population, oer a long time, te population will conerge to te equilibrium points as sown in Fig 2-3.

8 111 Kingdao Cangklang et al, 216 Australian Journal of Basic Applied Sciences, 1(11) Special 216, Pages: ) Te iger of te infection from mosquitoes to umans te infection from people to mosquitoes ( ) ( ) increase te infected indiiduals as sown in Fig 3. t concluded tat if te infected mosquitoes is ig, te number of dengue infection will be increase. ACKNOWLEDGMENTS Surapol Naowarat would like to tank Department of Matematics, Faculty of Science Tecnology, Suratani Rajabat Uniersity for equipment support te anonymous reiewers for teir kind elpful comments. REFERENCES Allen, L.J.S., 26. An ntroduction to Matematical Biology. Prentice Hall, New York. Batt, S., P.W. Geting, O.J. Brady, J.P. Messina, A.W. Farlow, C.L. Moyes, Te global distribution burden of dengue. Nature, 496: Brady, O.J., P.W. Geting, S. Batt, J.P. Messina, J.S. Brownstein, A.G. Hoen, 212. Refining te global spatial limits of dengue irus transmission by eidence-based consensus. PLoS Negl Trop Dis., 6: e176. CDC., dengue. Cutima, K., 215. Dynamics Model of H-Foot-Mout Disease wit Effect of H Wasing Campaign. Australian Journal of Basic Applied Sciences, 9(13): Guoyu Niu, Dengue irus enelope domain pro base on a tetraalent antigen secreted from insect cells:potential use for serological diagnosis. Najot, M., 214. Modeling Analysis of an SRS Epidemic Model wit Effect of Awareness Programs by Media. nternational Journal of Matematical, Computational, Natural Pysical Engineering, 8(1): Van den Driessce, P., James Watmoug, 22. Reproduction numbers sub tresold Endemic equilibriums for compartmental models of disease transmission. Matematical Biosciences, 18: Samita Adikari, (4): Burden estimation of dengue at National Public Heait Laboratory,Katmu. Jeelani, S., S. Sabesan, S. Subramanian, 215. Community knowledge,awareness preentie practices regarding dengue feer in Puducerry-Sout ndia. Suksawat Naowarat, 214. Effect of Education Campaign on Transmission Model of Conjunctiitis. Australian Journal of Basic Applied Sciences, 9(7): WHO.,